Mixtures

#thermo
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suppose we mix 3kg of water and 1kg of pure H2SO4 at 21.1 C adiabatically, what will the final temperature be?
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Info

if we actually did this isothermally, we'd have to remove energy since the total enthalpy of the mixture reduces. This is because of hydrogen bonding formed between h2so4 and h20

The above equation can be used for any of the variables like gibbs free energy, entropy, etc

Redlich Kister expansion

This is a polynomial fitted for delta H mix
x1 and x2 are the mole fractions of the two components
Todo

Fit a redlich kister expansion to H2SO4

Why are we doing this?

One of the things that separates chemical engineers from the rest of humanity is that we are able to separate stuff.
Distillation, crystallization, liquid liquid extraction, etc.
So we'll have a mixture and a process which outputs a number of pure components
Now if we want to know the minimum work or heat required to separate something, we can use this

Equilibrium

Imagine we have a phase equilibrium between a liquid and it's vapour:
I don't know why you should imagine this I just copied this off the board
The above can be written for vapour as well
the derivative of G with respect to components in the mixture is called the chemical potential of that particular component in whichever phase it's in
This is also called the partial molar gibbs energy
An interesting derivation later we see that at equilibrium, the chemical potential of each component in the liquid phase must be equal to the chemical potential of that component in the vapour phase

Relationship between and partial molar gibbs energy, and other partial molar quantities with their Del mix values

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Last year we had learnt that enthalpy is the addition of enthalpy multiplied by the mole fraction of each component, THIS WAS WRONG
It's the partial molar enthalpy of a component multiplied by it's mole fraction and added up with all the other components - thats the actual enthalpy
Gibbs duhem - get familiar

Partial molar characteristics

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Now, using the Redlich Kister expansion
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Special case: Mixture of ideal gases

N number of ideal gases are being mixed at a certain temperature
Volume change due to mixing is 0,
This means that the partial molar volume is equal to the volume of the pure substance
Problem

Suppose we have an isothermal separation like:

Solution

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Condition for Equilibrium

The Partial Molar gibbs free energy of all components has to be the same
Problem

A problem from the green envelope

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Bdababsoia eiasid
The del mix when x tends to 0 also ends up tending to 0

Todo

I need to study sandler later today after the graph theory and all ka kaam

Danger

Another problem. I am so hungry I don't think I can solve it I shouldn't have skipped breakfast

Non ideal mixture -
Mex is the excess property of a real gas

Important

A reminder for why we are doing all this - we are doing this to get to raoults law, henry's law and solubility
You might notice a pattern here, but that's because you are stupid and actually for quantities like entropy and gibbs free energy and the sort, there is another term after the delta mix to get the excess

Another way to write excess properties

Now, using summability relation
Now if we want the partial molar excess property (what a mouthful) at constant temperature, pressure and other components' moles:
Now wat dis look lyk?

Messing around with residuals

The above is for one component. The below is for multiple components
The f_i bar is the fugacity of the component in the mixture, NOT the partial molar fugacity
Pyi is the partial pressure of the gas
=>

A little further?

At equilibrium:
the above only works for ideal mixtures - like the fi bar related to the pure component fugacity

for the non ideal condition:
the a_i is the activity coefficient

Quote

If Prathamesh were normal, his activity coefficient would be 1/88, but since he interacts with 7 people in the class at any point in time, his activity coefficient is 7/88

for vapour phase, IM:
phi is the fugacity coefficient, which is the ratio of fugacity to the actual pressure
since the vapour phase is an IM, it's pressure is low enough and the pressure is equal to the fugacity
the exponent to e is small enough for that part of the equation to be equal to 1
If liquid phase is also an ideal mixture then,
Dis is raoults law